Question

In the circuit shown in the figure (32-E13), ε₁ = 3 V, ε₂ = 2 V, ε_a = 1 V and r₁ = r₂ = r₃ = 1Ω. Find the potential difference between the points A and B and the current through each branch.

Answer 1

Applying KVL in loop 1, we get:
$$\begin{gathered} i_1+\left(i_1-i_2\right)-1+2=0 \\ 2i_1-i_2=-1...\left(1\right) \end{gathered}$$
Similarly, for loop 2:
$$\begin{gathered} i_2-\left(i_1-i_2\right)-2+3=0 \\ 2i_2-i_1=-1...\left(2\right) \end{gathered}$$
Solving (1) and (2), we get:
$i_1=i_2=1\mathrm{A}$
Potential difference between A and B:
$V_{\mathrm{A}}-V_{\mathrm{B}}=E_2-\left(i_1-i_2\right)\times 1=2-\left(1-1\right)=2\mathrm{V}$
Current through the top branch = i₂ = 1 A
Current through the middle branch = i₁ - i₂ = 0 A
Current through the bottom branch = i₁ = 1 A